# Original Solution of Coupled Nonlinear Schrödinger Equations for Simulation of Ultrashort Optical Pulse Propagation in a Birefringent Fiber

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Coupled Nonlinear Schrödinger Equations System

_{i}—complex envelopes of the optical impulse of the i-th mode, α

_{i}—attenuation coefficient of the i-th mode; β

_{1,i}, β

_{2,i}, β

_{3,i}—the first, second and third order dispersion parameters of the i-th mode respectively; γ

_{i}—nonlinearity parameter for the i-th mode; C

_{i}

_{,m}, B

_{i}

_{,m}—coupling coefficients between the i-th and m-th modes; T

_{R}—Raman scattering parameter; ω

_{0,i}—angular frequency of the i-th mode; z—coordinate along the axis of the fiber; t—time.

_{X}and A

_{Y}), propagating in a birefringent fiber, which is used for modeling of short optical pulses propagation, has a form equivalent to Equation (1):

## 3. Initial Conditions and Boundary Terms

## 4. Dimensionless Equations

^{½}m

^{−1}], hence the dimensions of the equations constants are:

_{a}—characteristic length, T

_{a}—time, and P

_{a}—power:

## 5. The Finite-Difference Scheme

_{x}(x, y), F

_{y}(x, y) are used to denote appropriate parts in Equation (9).

_{1x}(x, y), Φ

_{1x}(x, y), Φ

_{2y}(x, y), Φ

_{2y}(x, y), and a third-order partial derivative by time, do not allow the use of the classic approach to equation system solutions (Thomas, or tridiagonal matrix, algorithm). The modification of Crank–Nicolson computing scheme is offered in [46]. The main idea is to write all linear terms in implicit form, and all nonlinear terms in explicit form. Using the recommendations given in [46], we write the expressions for F

_{x}(x, y) and F

_{y}(x, y) as the sum of linear and nonlinear parts:

_{1x}(x, y), Φ

_{2x}(x, y), Φ

_{1y}(x, y), Φ

_{2y}(x, y), while for the linear parts of Equation (16), the implicit form will be used.

_{1x}(x, y) and Φ

_{1y}(x, y) can be written as:

_{2x}(x, y) and Φ

_{2y}(x, y):

## 6. Boundary Conditions

## 7. The Line Equation System in Classic Form

**M**is four-diagonal matrix, where in addition to main diagonal, which consists of “C” coefficients, contains one “upper” (“D”) and two “sub” (“A” and “B”) diagonals. There is no need to store the whole matrix

**M**in computer memory. Instead, we use the two sets of five arrays for all four diagonals and its right member vector for each part of Equation (23). The

**A**and

**B**arrays are used for two sub diagonals storage,

**C**—for main diagonal,

**D**—for upper diagonal and

**R**—for right member vector:

**A**,

**B**,

**C**, and

**D**, used for matrix

**M**elements notation.

**M**to the triangular form, according to:

**M**(with right member vector

**R**) is transformed, we calculate unknown vectors according to:

**T**.

## 8. Numerical Solution Refining Algorithm

## 9. Method Verification on Some Classic Tasks

#### 9.1. Heat Diffusion in a Solid Rod Task

_{1,x}= β

_{1,y}= 0, β

_{2,x}≠ 0, β

_{2,y}= 0, β

_{3,y}= β

_{3,y}= 0, γ

_{x}= γ

_{y}= 0.

_{2,x}≠ 0. If we also request the β

_{1,x}≠ 0, the heat convection appears.

_{1,x}≠ 0, β

_{1,y}= 0, β

_{2,x}≠ 0, β

_{2,y}= 0, β

_{3,y}= 0, β

_{3,y}= 0, γ

_{x}= 0, γ

_{y}= 0 are shown in Figure 2b. We can see, that in process of time (with z growing), in addition to previous effects, the heat point is moving along the length (along the t variable). The heat diffusion task solution, based on coupled nonlinear Schrödinger equations system, demonstrates an excellent matching with classic heat diffusion equation solutions, including analytical solutions, by its character and values.

#### 9.2. The Korteweg–De Vries and Linear Tasks

_{1,x}= 0, β

_{1,y}= 0, β

_{2,x}= 0, β

_{2,y}= 0, β

_{3,y}≠ 0, β

_{3,y}= 0, γ

_{y}= 0 and choose special and different values γ

_{x}for individual nonlinear terms, we receive the Korteweg–de Vries equation. The obtained computing results are presented in Figure 3a. Its common character almost coincides with our previous results [29,30,31,46], and other authors’ results [48].

#### 9.3. The Ultra-Short Pulse Evolution in Fiber

_{3,y}= β

_{3,y}= 0, T

_{R}→ 0 and ω

_{0}→ ∞. The computing results of the Manakov equations system task, received on the base of coupled nonlinear Schrödinger equations system, is shown in Figure 4a. These results coincide with our previous results [29,30,31,46] and other researchers‘ results [23,24,25,26,27,28,49].

_{1,x}= 4.294 × 10

^{−9}s/m, β

_{1,y}= 4.290 × 10

^{−9}s/m, β

_{2,x}= 3.600 × 10

^{−26}s

^{2}/m, β

_{2,y}= 3.250 × 10

^{−26}s

^{2}/m, β

_{3,y}= β

_{3,y}= 2.750 × 10

^{−41}s

^{3}/m, γ

_{x}= γ

_{y}= 3.600 × 10

^{−2}(m⋅W)

^{−1}, T

_{R}= 4.000 × 10

^{−15}s, ω

_{0}= 2.3612 × 10

^{−15}s

^{−1}(wavelength 798 nm). The single chirped Gauss pulse is in the input fiber end (chirp C = –0.4579), pulse duration is 12 fs, with maximum power P = 1.75 × 10

^{5}W. The pulse form is described as:

^{−4}d.u. Besides, the automatic integration step correction algorithm was included. It allowed the calculation of pulse evolution length up to ~2.5 mm. The maximum error for iteration process was chosen as 10

^{−30}d.u. All calculations were made in a processor with double precision and 64-bit architecture.

## 10. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Eight-point finite-difference computing scheme for coupled nonlinear Schrödinger equations system integration at: (

**a**) internal area; (

**b**) boundary.

**Figure 2.**Heat diffusion in the solid rod task numerical results: (

**a**) without convection; (

**b**) with connectivity. Curves for different length values are marked: red at z = 0 (initial), brown at z = 1/3, green at z = 2.3, blue at z = 1.

**Figure 3.**The numerical results of: (

**a**) Korteweg–de Vries equation; (

**b**) coupled linear Schrödinger equations system. Curves for different length values are marked: red at z = 0 (initial), brown at z = 1/3, green at z = 2.3, blue at z = 1.

**Figure 4.**The pulse form evolution in fiber of: (

**a**) Manakov equations system; (

**b**) coupled nonlinear Schrödinger equations system. Curves for different length values are marked: red at z = 0 (initial), brown at z = 1/3, green at z = 2.3, blue at z = 1.

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**MDPI and ACS Style**

Zhavdatovich Sakhabutdinov, A.; Ivanovich Anfinogentov, V.; Gennadievich Morozov, O.; Alexandrovich Burdin, V.; Vladimirovich Bourdine, A.; Mudarrisovich Gabdulkhakov, I.; Anatolievich Kuznetsov, A.
Original Solution of Coupled Nonlinear Schrödinger Equations for Simulation of Ultrashort Optical Pulse Propagation in a Birefringent Fiber. *Fibers* **2020**, *8*, 34.
https://doi.org/10.3390/fib8060034

**AMA Style**

Zhavdatovich Sakhabutdinov A, Ivanovich Anfinogentov V, Gennadievich Morozov O, Alexandrovich Burdin V, Vladimirovich Bourdine A, Mudarrisovich Gabdulkhakov I, Anatolievich Kuznetsov A.
Original Solution of Coupled Nonlinear Schrödinger Equations for Simulation of Ultrashort Optical Pulse Propagation in a Birefringent Fiber. *Fibers*. 2020; 8(6):34.
https://doi.org/10.3390/fib8060034

**Chicago/Turabian Style**

Zhavdatovich Sakhabutdinov, Airat, Vladimir Ivanovich Anfinogentov, Oleg Gennadievich Morozov, Vladimir Alexandrovich Burdin, Anton Vladimirovich Bourdine, Ildaris Mudarrisovich Gabdulkhakov, and Artem Anatolievich Kuznetsov.
2020. "Original Solution of Coupled Nonlinear Schrödinger Equations for Simulation of Ultrashort Optical Pulse Propagation in a Birefringent Fiber" *Fibers* 8, no. 6: 34.
https://doi.org/10.3390/fib8060034